Exercise 1.1.7

Question

Describe the column space of $A=\left[\begin{array}{lll}\boldsymbol{v} & \boldsymbol{w} & \boldsymbol{v}+2 \boldsymbol{w}\end{array}\right]$. Describe the nullspace of $A:$ all vectors $\boldsymbol{x}=\left(x_{1}, x_{2}, x_{3}\right)$ that solve $A \boldsymbol{x}=\mathbf{0}$. Add the "dimensions" of that plane (the column space of $A$) and that line (the nullspace of $A$): dimension of column space $+$ dimension of nullspace $=$ number of columns

Neil Z. · Accepted Answer

$A=\begin{bmatrix}v & w & v + 2w\end{bmatrix}$, the column space is a plane defined by the combination of vectors $v$ and $w$, i.e. $cv+dw$.\par

Now compute the nullspace of $A$, we let $Ax=0$, that is

\begin{align*}
Ax &= \begin{bmatrix}v & w & v + 2w\end{bmatrix}\begin{bmatrix}x_1\x_2\x_3\end{bmatrix}\
&= x_1v + x_2 w + x_3 (v+2w)\
&= (x_1 + x_3)v + (x_2 + 2x_3)w\
&= 0
\end{align*}

The solution is thus $x=\begin{bmatrix}x_1\2x_1\-x_1\end{bmatrix}$ and the nullspace of $A$ is the line defined by $x$. \par

It's easy to see that the dimension of column space + dimension of null space = number of columns in $A$.

Exercise 1.1.7

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Exercise 1.1.7

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